Lemma 67.34.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ Let $n \geq 0$. Assume $f$ is locally of finite presentation. The open

of Lemma 67.34.4 is retrocompact in $|X|$. (See Topology, Definition 5.12.1.)

Lemma 67.34.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ Let $n \geq 0$. Assume $f$ is locally of finite presentation. The open

\[ W_ n = \{ x \in |X| \text{ such that the relative dimension of }f\text{ at } x \leq n\} \]

of Lemma 67.34.4 is retrocompact in $|X|$. (See Topology, Definition 5.12.1.)

**Proof.**
Choose a diagram

\[ \xymatrix{ U \ar[r]_ h \ar[d]_ a & V \ar[d] \\ X \ar[r] & Y } \]

where $U$ and $V$ are schemes and the vertical arrows are surjective and étale, see Spaces, Lemma 65.11.6. In the proof of Lemma 67.34.4 we have seen that $a^{-1}(W_ n) = U_ n$ is the corresponding set for the morphism $h$. By Morphisms, Lemma 29.28.6 we see that $U_ n$ is retrocompact in $U$. The lemma follows by definition of the topology on $|X|$, compare with Properties of Spaces, Lemma 66.5.5 and its proof. $\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)